Multilevel Richardson-Romberg extrapolation

Abstract

We propose and analyze a Multilevel Richardson-Romberg (MLRR) estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg method introduced in [Pa07] and the variance control resulting from the stratification introduced in the Multilevel Monte Carlo (MLMC) method (see [Hei01, Gi08]). Thus, in standard frameworks like discretization schemes of diffusion processes, the root mean squared error (RMSE) $\epsilon >0$ can be achieved with our MLRR estimator with a global complexity of ${\epsilon }^{-2}\log (1/\epsilon )$ instead of ${\epsilon }^{-2}(\log (1/\epsilon ){)}^2$ with the standard MLMC method, at least when the weak error $\mathbf{E}[Y_h]-\mathbf{E}[Y_0]$ of the biased implemented estimator $Y_h$ can be expanded at any order in $h$ and $\Vert Y_h-Y_0{\Vert }_2=O(h^{\frac{1}{2}})$. The MLRR estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error $\Vert Y_h-Y_0{\Vert }_2=O(h^{\frac{\beta }{2}})$, $\beta <1$, the gain of MLRR over MLMC becomes even more striking. We carry out numerical simulations to compare these estimators in two settings: vanilla and path-dependent option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.

38 pages

References in corpus (4)

• Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction
• Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
• Central limit theorem for the multilevel Monte Carlo Euler method
• Multilevel path simulation for weak approximation schemes

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• Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests
• Multilevel Importance Sampling for Rare Events Associated With the McKean--Vlasov Equation
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• Asymptotics for the normalized error of the Ninomiya-Victoir scheme
• Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems
• Improving MLMC for SDEs with application to the Langevin equation
• Thinning and Multilevel Monte Carlo for Piecewise Deterministic (Markov) Processes. Application to a stochastic Morris-Lecar model